06_Lecture_I_03_Elimination.ipynb

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    "+ This notebook is part of the addition lecture *An overview of key ideas* in the OCW MIT course 18.06 by Prof Gilbert Strang [1]\n",
    "+ Created by me, Dr Juan H Klopper\n",
    "    + Specialist Surgeon\n",
    "    + Groote Schuur Hospital\n",
    "    + Senior Lecturer in Surgery and Biostatistics & Machine Learning\n",
    "    + University Cape Town\n",
    "    + <a href=\"mailto:juan.klopper@uct.ac.za\">Email me with your thoughts, comments, suggestions and corrections</a> \n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\"><img alt=\"Creative Commons Licence\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" href=\"http://purl.org/dc/dcmitype/InteractiveResource\" property=\"dct:title\" rel=\"dct:type\">Linear Algebra OCW MIT18.06</span> <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">IPython notebook [2] study notes by Dr Juan H Klopper</span> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\">Creative Commons Attribution-NonCommercial 4.0 International License</a>.\n",
    "\n",
    "+ [1] <a href=\"http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/index.htm\">OCW MIT 18.06</a>\n",
    "+ [2] Fernando Pérez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org"
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   "source": [
    "### Library import"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import init_printing, Matrix, symbols, eye, Rational\n",
    "init_printing()"
   ]
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   "metadata": {},
   "source": [
    "# Elimination"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A system of linear equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the previous lesson, we had a brief glimpse at linear systems.  The _linear_ in _linear systems_ refers to the fact that each variable appears on its own (i.e. to the power $1$ and not in the form $x \\times y$ or the like) and it is not transcendental.  If a solution exists, it then satisfies all of the equations at once.  We will consider the linear system in (1)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 1x+2y+1z &= 2 \\\\ 3x + 8y + 1z &= 12 \\\\ 0x + 4y + 1z &= 2 \\end{align} \\tag{1} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A possible solution for $x,y$, and $z$ is given in (2), where $x=2$, $y=1$, and $z = -2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 1\\left(2\\right)+2\\left(1\\right)+1\\left(-2\\right) &= 2 \\\\ 3\\left(2\\right)+8\\left(1\\right)+1\\left(-2\\right) &= 12 \\\\ 0\\left(2\\right)+4\\left(1\\right)+1\\left(-2\\right) &= 2 \\end{align} \\tag{2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since (1) is a set (three) equations that have a solution ( or possibly solutions) for their variables in common, all left- and all right hand sides can be manipulated in certain ways."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could simply exchange the order of the equations.  In (3) the second and third equations have been exchanged, called _row exchange_."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align}1x+2y+1z &= 2 \\\\ 0x + 4y + 1z &= 2 \\\\ 3x + 8y + 1z &= 12 \\end{align} \\tag{3} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could multiply both the left- and right-hand side of one of the equations with a scalar.  In (4) we multiply the first equation by $2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 2x+4y+2z &= 4 \\\\ 3x + 8y + 1z &= 12 \\\\ 0x + 4y + 1z &= 2 \\end{align} \\tag{4}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lastly, we can subtract a constant multiple of one equation from another."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These three _manipulations_ serve an excellent purpose, as it allows us to eliminate of one (or more) of the variables (that is, give it a coefficient of $0$).  Remember that we are trying to solve for all three equations and have three unknowns.  We can most definitely struggle by doing this problem algebraically by substitution, but linear algebra makes it much easier."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In (5) we have multiplied the first equation by $3$ (both sides, so that we maintain integrity of the equation) and subtracted the left hand side of this new equation from the left-hand side of the second equation and the new right-hand side of the first equation from the right-hand side of the second equation.  This is quite legitimate, as the left- and right-hand sides are equal (it is an equation after all) and so, when subtracting from the second equation, we are still doing the same thing to the left-hand side as the right-hand side."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 1x+2y+1z &= 2 \\\\ 0x + 2y - 2z &= 6 \\\\ 0x + 4y + 1z &= 2 \\end{align} \\tag{5} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This has introduced a nice $0$ in the second equation.  Let's go further and multiply the second equation by $2$ and subtract that from the third equation as seen in (6) below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 1x+2y+1z &= 2 \\\\ 0x + 2y - 2z &= 6 \\\\ 0x + 0y + 5z &= -10 \\end{align} \\tag{6} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let last equation is easy to solve for $z$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ z=-2 \\tag{7}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Knowing this, we can go back up to the second equation and solve for $y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 2y+2(-2) &= 6 \\\\ y &= 1 \\end{align} \\tag{8} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, up to the first equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} x+2(1)+1(-2) &= 2 \\\\ x &= 2 \\end{align} \\tag{9} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "W have solve the linear system by substitution.  We need to have gone straight for substitution, though.  Indeed, we could have tried to get zeros above all our leading (non-zero) coefficients.  Let's just clean up the third equation by multiplying throughout by $\\frac{1}{5}$ as in (10) below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 1x+2y+1z &= 2 \\\\ 0x + 2y - 2z &= 6 \\\\ 0x + 0y + 1z &= -2 \\end{align} \\tag{10} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we have to get rid of the $-2z$ in the second equation, which we can do by multiplying the third equation by $-2$ and subtracting from the second equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align}  1x+2y+1z &= 2 \\\\ 0x + 2y - 0z &= 2 \\\\ 0x + 0y + 1z &= -2 \\end{align} \\tag{11}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiplying the second equation by $\\frac{1}{2}$ yields (12)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align}1x+2y+1z &= 2 \\\\ 0x + 1y + 0z &= 1 \\\\ 0x + 0y + 1z &= -2 \\end{align} \\tag{12} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can do the same to get rid of the $1z$ in the first equation (multiply the third equation by $1$ and subtract it from the first equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 1x+2y+0z &= 4 \\\\ 0x + 1y + 0z &= 1 \\\\ 0x + 0y + 1z &= -2 \\end{align} \\tag{12}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now tow get rid of the $2y$ in the first equation, which is above our leading $1y$ in the second equation.  Simple enough, we multiply he second equation by $2$ and subtract that from the first equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} 1x+0y+0z &= 2 \\\\ 0x + 1y + 0z &= 1 \\\\ 0x + 0y + 1z &= -2 \\end{align} \\tag{13} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution is now clear for $x,y$, and $z$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need not rewrite all of the variables all the time.  We can simply write the coefficients.  The augmented matrix of coefficients is in (14)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{bmatrix} 1&2&1&2\\\\3&8&1&12\\\\0&4&1&2 \\end{bmatrix} \\tag{14} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A matrix has rows and columns (attached, in position, to our algebraic equation above).  We simply omit the variables.  The left-upper entry is called the pivot.  Our aim is to get everything below it to be a zero (as we did with the algebra).  We do exactly the same as we did above, which is multiply row 1 by 3 and subtract these new values from row 2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{bmatrix} 1&2&1&2\\\\0&2&-2&6\\\\0&4&1&2 \\end{bmatrix} \\tag{15} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now $2$ times row 2 subtracted from row 3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{bmatrix} 1&2&1&2\\\\0&2&-2&6\\\\0&0&5&-10 \\end{bmatrix} \\tag{16} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply the last row with $\\frac{1}{5}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{bmatrix} 1&2&1&2\\\\0&2&-2&6\\\\0&0&1&-2 \\end{bmatrix} \\tag{17} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This shows $z = -2$ in the last row of (17).  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With this small matrix, it's easy to do back substitution as we did algebraically.  The first non-zero number in each row is the pivot (just like the upper-left entry).  The steps we have taken up to this point is called _Gauss elimination_ and the form we end up with is _row-echelon form_.  We could carry on and do the same sort of thing to get rid of all the non-zero entries above each pivot.  This is called _Gauss-Jordan elimination_ and the result is _reduced row-echelon form_ (see the computer code below)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All of these steps are called _elementary row operations_.  The only one we didn't do is _row exchange_.  We reserve this action so as not to have leading (in the pivot position) zeros."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's create some code to show-case elementary row operations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
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    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 1 & 2\\\\3 & 8 & 1 & 12\\\\0 & 4 & 1 & 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2  1  2 ⎤\n",
       "⎢           ⎥\n",
       "⎢3  8  1  12⎥\n",
       "⎢           ⎥\n",
       "⎣0  4  1  2 ⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_augmented = Matrix([[1, 2, 1, 2], [3, 8, 1, 12], [0, 4, 1, 2]])\n",
    "A_augmented"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can ask `sympy` to simply get the augmented matrix in reduced row-echelon form and read off the solutions.  This is done with the `.rref()` method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 0 & 2\\\\0 & 1 & 0 & 1\\\\0 & 0 & 1 & -2\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  2\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  0  2 ⎤           ⎞\n",
       "⎜⎢           ⎥           ⎟\n",
       "⎜⎢0  1  0  1 ⎥, (0, 1, 2)⎟\n",
       "⎜⎢           ⎥           ⎟\n",
       "⎝⎣0  0  1  -2⎦           ⎠"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_augmented.rref() # The rref() method returns the reduced row-echelon form"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Elimination matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matrices can only be multiplied by each other if in order we have the first column size equal the second row size.  Rows are usually called $m$ and columns $n$ when considering their dimensions.  So, our augmented matrix above will be $m \\times n = 3 \\times 4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at how matrices are multiplied by looking at two small matrices in (18)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{bmatrix} {a}_{11}&{a}_{12} \\\\ {a}_{21}&{a}_{22} \\end{bmatrix} \\\\ \\\\ \\begin{bmatrix} {b}_{11}&{b}_{12}\\\\{b}_{21}&{b}_{22} \\end{bmatrix} \\tag{18} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The subscripts refer to row and column position, i.e. $21$ means row $2$ column $1$>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that we have a two $ 2 \\times 2 $ matrices.  The *inner* two values are the same ($2$ and $2$), so this multiplication is allowed.  The resultant matrix will have the size equal to the *outer* two values (first row and last columns); here also a $2 \\times 2$ matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So let's look at position $11$ (row $1$ and column $1$).  To get this we take the entries in row $1$ of the first matrix and multiply them by the entries in the first column of the second matrix.  We do this element by element and add the multiplication of each set of separate elements to each other.  The python code below shows you exactly how this is done."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "a11, a12, a21, a22, b11, b12, b21, b22 = symbols('a11 a12 a21 a22 b11 b12 b21 b22')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}a_{11} & a_{12}\\\\a_{21} & a_{22}\\end{matrix}\\right], \\  \\left[\\begin{matrix}b_{11} & b_{12}\\\\b_{21} & b_{22}\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡a₁₁  a₁₂⎤  ⎡b₁₁  b₁₂⎤⎞\n",
       "⎜⎢        ⎥, ⎢        ⎥⎟\n",
       "⎝⎣a₂₁  a₂₂⎦  ⎣b₂₁  b₂₂⎦⎠"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[a11, a12], [a21, a22]])\n",
    "B = Matrix([[b11, b12], [b21, b22]])\n",
    "A, B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATAAAAAyCAYAAADbRIdMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAKpUlEQVR4Ae2d7ZEUNxCGlysCoHAEPjIwdgSGDLAdASYDKH7BP8pkAGTAOQO4CPjIwIQAG4Lfd260aLWSurWrj5mjVTU7GrWm1Xqk7dVoZ0Y3nj179m2z2dzC5oeXz58/f+InWNwIGAEjMIoA/NE/KPtxUP725pzwB/afPeFXL25RI2AEjMBoAi9gwCvPiF8Qf+Mc2Bd4uC+e0KJGwAgYgcUQgH/awhhuU8DxdNXoHJhLH76fDbuAIb9i+4rjO8ONWpEBxq+ssYxXGa8w92h+Z6FBo48BZIvtPuzgiPD9aHvWVr7xK2sx41XGK8w9mt/iHJgHiNe477zjYVE00gNs4QTiMHuUBS+Gn9Le0dkWw8v6m74rLNKBoQHvzVWwEZi+LXc5jd8OhSpivFSYkplG8lukAwOp6RISYHaTdkl6JogRMH4xKuk045Vmo5EM46eaxIcjOUcteF8YHQpn///D9hTbz42cDEdgn6HbXbZxIv8djv/FfnXB+JU1mfEq4xXL3ZnhsO+rOAIDCBr3Cds/iD/B9gjxv7DdQrz6CAk66SA5H0Gn+RrHL7Gn87xAnGmrCrDZ+BW0mPEqgJXI2pMhyhr6fb2ZYDAlz8bxlgY6Lv8+sY9Ii97sinx0PnR2HFYeBEmOE/iFZ/gdeScHyT02pj3ARoc2BUmXJHd6Wu1RPhu3Gj/o493I1ElHzvZgu0yMEHdBxU+jC3mybekKrLVHeVV50S5FHVS8JF0oR9M2tVAl9dRmqKiXip+kR5KnKnwzJZjTXaO8DfLRaP+u2A0MYGfn6IzhYKQkya9Omz7p+Hj5uPtiIs6OzfATPyRdkpw6YgHnsb6uQfwsLP825Bx5hoE3AfNJhlioyY+6XqGs6YcEezrGS2x3g4I1/LK6oDvblkF5NQ9r8tLWQcMrqwu8sjxTgObzavY3FlWToaZeGn5ZPcfyY2UlB/Yn8rxHAaEzoYPa+4cQefgo0iPs2SAHjSLJcY4LvIGVIzw/MI3hAz8kXZKcOmIB5/FS9SAgnSO/c+x3o7+DTPGEavygnjb4Pxp8tOITbKJd/uhY5Cfpgr5sW+L8VqEar4I6iLwUurRts8cNemv3N+qvxhC6NPUS+Sn0aMph3Q7C2UHKnAC4HHVwY2f2w+Sc5kb102vF+WvHOTc/uBHOntP0Mywt3oifP7J1Pyp+GjFo+fnnpXR1w9qIl8Z+LS9J13CejRhK9dLyk/RI8ih/aQTGk6ZRj3c2L6MmpwZg9Jy83PNHAF7Wo6M7fXOj8FeFozv3RTta8YATq/BD3cNHqthxGMLRKtOy/Ap1UV/PUIVXocFZXpKuBfKswrCgXll+kh5JnuN/lhJCKZ0FRzw7z4i0v3HML44z+DekuTiSqwTeKrErE/ELbG9Rzusq2jsp6cCP8wqxSfxj+KV0daI1TQtcp/42hOegPteiv6n5SSMwXrq9AZjHc0+mQ+OkHf9lZFqLR30eevo5ac/bJ1blvGCzC034gQcbmHOTsTm5In6CLlePXvsmvATji3gJujYL4NmEYaZeRfwyeia0kjzkn3VgUMZfRQIJQywtzHPU8Vym+zfzKB1LOakFP+jkKJj34EUZlfCTdPXmONse61uxtCrmlfCSClwCzxYMc/Uq4ZfTQ7aSPMY/68BiJ/ygaYv48wANfA/872A/Oa/5mK8cCv9oEZuppi6xsB8gQ2Wei+hvbLZa9ZL0SPJUF2rhwG6lCpvTJblw+p5Y0iXJ95SlDgB3m5I1SI/aDBs498hLxxeI888TBjqy4tFJga6oLVPJy/mQbJTkJTWJ6irgqSoL+raqjPUyNa2XxEeS56p5Y34nPu96L/4V9xXjfE68874WjhIY5+TeB6RP8zSSHHnVQdIlydUFdcwo2Qx5bO2CDdJvlJop6YI825al5bXIL9koyUtsknRBXq1tSuw6NW+vekl8JHmsnjiHP+iX1RxYrBBLMwJGwAi0IOAc2FkL5abTCBgBI9CDgDmwHpStDCNgBJoQMAfWBKspNQJGoAcBc2A9KFsZRsAINCFgDqwJVlNqBIxADwLmwHpQtjKMgBFoQqDFjawnGYq/R3lTHR/g5nuGeJd5+BaGk/Rf95ONX1kLG68yXmHu0fwWNwIDEFvYNuwlBcfGrwAWshqvMl5h7tH8FufAPEC807bF2y68InRRNJItbKtDteZc1t9Oa70h/BbpwOAw+DgSw2Iear0yZx2fxq+snYxXGa8w90h+i3RgAHQfGxfL6P1Qa9g2az02fmUtZ7zKeIW5h/FTTeLDkZzDYj6oTYfCSXZb2BYQtMH4aUld5TNeZbxiuTsz5BUTXy3vXnzabSFq0YHBKBrHfwXvIj69Php7LrpxsLAt0vm6Fzo4OjzmPXjlMfLwWjm3biTPZx6Gh8jPSX2mfcOe78LavcIacUlXVj6V0PgDNlbjB10avip+Sl3d+dXkxaaFvmwdIFfxknRBj9g2jbvaTj1s6dbntPwkPpJ8V7kgknVgs3F0XuLCtrMByXULIWdHcm8RpYNLBcJnSC5sK+mS5Ffq23/CDn45avHjFyTJ16uNhl9W1yh+lXn17G9Znl7bNI9WZqipV43+piknyi7rwHAGFfNLKC5sizzZtd0Alu8bS64bCZkLvJ7mcHTrEuZG4aFb2Darq6AsV8S0x3msr2sQX0YGTRe2hf4sP4Xc2Svyk3Qdy88ZcMJ+lf1N4pni0aC/sahqDJX1Orm/Kcth3Q6C5MC4nJlqYdtZs7t05KFzQH7anC274w2sH4McTGP4cLVr84kOVXuh0dr8fJYpvlp+Gl1tQKe11uaVLum7RMvr+xnxWDHPBv2NltVmKNVLy0/SI8mj1JMODHA56uAWvql1GqFAvpeO4/COeQ7hGUJndJWa/uR5/grUzOlem7ya2ypq8yvgK/Ir0JVupcqS2rwKzBN5SbqWwrM2Q2W9RH6SHkme43+WE86ycNSzt7AtCqfnjAUOZQ8m8WMZI2n+RD2dKH9VruXCtifwy/Et5ZfTFWmepknXob+N5tmKYapetftbqpyDjpd0YPhi8RKFI56dg0KaamFb5KMBqXULD4wIEo5ZKDNQMf6wJT+BbxE/QVc3kC15CZUo4iXo2ozk2ZJhpl5F/DJ6JrSSPOSfvIScM/LSrWhhWxiQXbcwNCBy/BBpbuHcH25hW4mfJAc7NT+FrkjzNE1adX9bCM/qDIV6VetvQjnRjpd1YFDIUZibf/IVxNI2yM/5sZPWLZzLdLdb+GWuLl6bn4avlp9GV2/gtXlp7NfyknQthWdthlK9tPwUeo7yHVkHJjWaL4eBnMzjpaO0biHntGoFSZck19rR/M8DiZ8k11aE+Qp01eJXYp4q76A6RHkU2KKqGzI17280RLJbkmsrI+mR5LlyqjkwFHKJjQ184RcI47Y8xp5zabt1I3HMfLt1I5lHGyRdklxbjssHfVMd3HGjfZYfypTkJWZlddXmV2JYQd5udVDwyNpSUKcpa6f+xrIkuyW5tmqSHkmeLMfWhUyiMYERMAJLJQAnzyu+y+S/kEs13OwyAkbACDgC5sAcCdsbASOwOgJuDuw8uO7mu+h7zPusDpgZbASMQH8C8EecX7/tlTzdn+oc2N7EOzK9xBZ9LtBTYFEjYASMQC8CT1HQ46Cw7f8yZCmRDVS4QwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22}\\\\a_{21} b_{11} + a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡a₁₁⋅b₁₁ + a₁₂⋅b₂₁  a₁₁⋅b₁₂ + a₁₂⋅b₂₂⎤\n",
       "⎢                                    ⎥\n",
       "⎣a₂₁⋅b₁₁ + a₂₂⋅b₂₁  a₂₁⋅b₁₂ + a₂₂⋅b₂₂⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's constrain ourselves to the matrix of coefficients (this discards the right-hand side from the augmented matrix above)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 1\\\\3 & 8 & 1\\\\0 & 4 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2  1⎤\n",
       "⎢       ⎥\n",
       "⎢3  8  1⎥\n",
       "⎢       ⎥\n",
       "⎣0  4  1⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, 2, 1], [3, 8, 1], [0, 4, 1]]) # I use the same computer variable above, which\n",
    "# will change its value in the computer memory\n",
    "A # A 3 by 3 matrix, which we call square"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The _identity matrix_ is akin to the number $1$, i.e. multiplying by it leaves everything unchanged.  It has ones along what is called the main diagonal and zeros everywhere else."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  1  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  0  1⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = eye(3) # Identity matrices are always square and the argument\n",
    "# here is 3, so it is a 3 by 3 matrix\n",
    "I # Note what the main diagonal is"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's multiply $I$ by $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 1\\\\3 & 8 & 1\\\\0 & 4 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2  1⎤\n",
       "⎢       ⎥\n",
       "⎢3  8  1⎥\n",
       "⎢       ⎥\n",
       "⎣0  4  1⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I * A # Nothing will change"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get rid of the leading $3$ in the second row(because we want a $0$ under the first pivot in the first row), we multiply the first row by $3$ and subtracted that from the second row.  Interestingly enough, we can do the same to the identity matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\-3 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1   0  0⎤\n",
       "⎢        ⎥\n",
       "⎢-3  1  0⎥\n",
       "⎢        ⎥\n",
       "⎣0   0  1⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E21 = Matrix([[1, 0, 0], [-3, 1, 0], [0, 0, 1]])\n",
    "E21 # 21 because we are working on row 2, column 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That gives us the required 3 times the first row and the negative shows that we subtract (add the negative).  It's a thing of beauty!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 1\\\\0 & 2 & -2\\\\0 & 4 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2  1 ⎤\n",
       "⎢        ⎥\n",
       "⎢0  2  -2⎥\n",
       "⎢        ⎥\n",
       "⎣0  4  1 ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E21 * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just what we wanted.  $E1$ is called the first elimination matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's do something to the identity matrix to get rid of the $4$ in the third row (the second column).  It would require $2$ times the second row subtracted from the third row.  Look carefully at the positions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & -2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0   0⎤\n",
       "⎢        ⎥\n",
       "⎢0  1   0⎥\n",
       "⎢        ⎥\n",
       "⎣0  -2  1⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E32 = Matrix([[1, 0, 0], [0, 1, 0], [0, -2, 1]])\n",
    "E32"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 1\\\\0 & 2 & -2\\\\0 & 0 & 5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2  1 ⎤\n",
       "⎢        ⎥\n",
       "⎢0  2  -2⎥\n",
       "⎢        ⎥\n",
       "⎣0  0  5 ⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E32 * (E21 * A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Spot on!  We now have nice pivots (leading non-zeros), with nothing under them (along the columns).  As a tip, try not to get fractions involved.  As far as the other two row operations are concerned, we can either exchange rows in the identity matrix or multiply the required row by a scalar constant."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Look at what happens when we multiply $E2$ and $E1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\-3 & 1 & 0\\\\6 & -2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1   0   0⎤\n",
       "⎢         ⎥\n",
       "⎢-3  1   0⎥\n",
       "⎢         ⎥\n",
       "⎣6   -2  1⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L_inv = E32 * E21\n",
    "L_inv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Later we'll call this matrix the inverse of $L$.  It is in triangular form, in this case lower triangular (note all the zeros above the main diagonal)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAABLCAYAAABz5qkHAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGfElEQVR4Ae2dX67UNhTGh6rPFS1S1dcOO7jtXQGwA9odUHYA6tO9bxXsAFgBhR0UVgDcHcBzVan0qhtov99MPPJk4sTOHMfJyJYy/js+x9/nf3FOZm5cXFycrVarD7q63OvLy8ufujJqmh0Cwvijalt31ai8G196Gc8VprDvPvmRGs6GwBPVfLNV+7ni90nzSXoi1iopLaSmiAp3BsieUxoEHZC0V2hOESlML6O34dy08FjpV9uk0/70R1JUSxuGX6jw9wpfR33piEINQS/k79ZGhSHsg/x7ut4cUb3ZV6UHvT4LLlEkSQF6MgowHdKT2/OnkrI5CHng1y59GEW/KO2Vrq/9vCnDU+ESSxIjZtOTG3A2c+VEgPwsOXd13W7JYwTdlz5rXUXWUsmdBJcvWg2fY3QzegVIaPSG0ufYllE6RY2kUTUbfUnk/BCoivs7XJFRtBU9zecSRtIBEiKO6Y+18Wkz5RyUOaWERZIkAp7p4jTk8SmREWrL4kgSMRD0Rv5uSx5q3KmkL4okEfMI4OU/PBUCYtqxGJJEDNv+2z5BCrP9Xsc0dMllFkGSiGAnd+4T1IA+5f1aMZ7HbMEnvS9pRgonC2wU3PkdOnyj60xpT4uhty84Gy7RJAkMFmyAYfuLe6u09/I5Qzs4xd2UsPmAIKa0zXrUqrL4PdIUuKSQVGSxFgihm9kWX2Wi0i87LotYk8rAPx+plaT5cBHUpJIUhGY+GZWk+XAR1KSSFIRmPhmVpPlwEdSkkhSEZj4ZlaT5cBHUpJIUhGY+GZWk+XAR1CT6WIgadATCAeffuq51Yb3zUmnZDRQlg8NLd7i6Vhi3CONIC92jR5KEYdT/Tj52BRyo/qbrlcIONEXtneqHIIwjHzbXPcXpGBzsusNee8EGNVrpHkWShGGIyAO21053hRlNxDmlzuk6jSMlEPm5ZR/bLhPdo0iSptgTdJnzvlM6z3To7bkcxpFdr+agz03JzjqSj2yUie6xJIWmFXozLpS/zT3uc8nGkSa6D24cvFHyuQfrbL1Z8kPPk2ZvHGmle8xI4mkszo2abWz72UecX840rMYzcukYizOOHKN7DEkO4L5155YrNJG/ZOPIZN0HpzuB3jda3Cjj3mkSp57ojCOzP7aWLDrm28SGPdD3uEU4cGN1h6SvdP3V+F0VX6ty0h0hfhk3uiYxCJEekxpHSh5TfGhN9HEYDI/Q/TtVCi8rprt/dX3b+PI63Wa725HjiOvanncUH5+kRi7WOHKk7n8KLXjZkBSDHDeNP3YU5GjoqulxHdk2Sar/TDUt0jjSQveYNWklQc91cVbGm3WbUwf5THWcRNyxoaK7FslhF0cnmbtx5EEDrHSPIqmRztz8ayOYpHNddxTvXCQpYOQgCKI261GrzknWwpbMlKiJ7tEkiQwW0cnfB5Jck4U7BVmrsla6p9wnWele60lEoJKUCFiJ4pWkEqgnyqwkJQJWonglqQTqiTIrSYmAlSheSSqBeqLMSlIiYCWKV5JKoJ4os5KUCFiJ4tHHQiinY44ixpGlZfcRI0ywZOJB5O8K8+yNE3seSP6h8M4ETvHRLnokNcpMbhxJy0rKjkB2rTKQ9I/0/E8+pEGWCUGqK+55kgQWM44sKRuAIhwn8fyWBKRwAM2vtpgeRMdOd33GkY+kFEaKnJLncCVlx7TnkzUpbaGx093d9hebuCMmlB/4WlJyqO4pZCcpmqvw4EhilDTC+6yGmJfNXUnZKY2RniwHOLDCpOCZ0swehsaMJGds4nouyjjXR5wrc4xfUnas3hDDzg4TA9YmdsCmb3zEkOSUdSPKxX0/t3FkSdl+Ow/CIobfJt91YIXZSGA9xY7PxMWQ1DdaXE/PZRxZUvYxAEOU2W/xDZLk9RJHiK+86+FZDEJKyvYbGQpLP16i476o7dzIWrczxsQHSWoqLWkcWVL2EKahnafrvO+HKojJjyWppHFkSdlDGLJh6LJmgjwzo9EokqQI78h+lr/7OU2F6S1sPff+R0JxU1dSdkRD+DujvQ2C4mDC0sBNuIkbvE/ypJQyjkSFkrI9CPaDIoTTBkcU6xBrEJsd03/EiSZJyqCE6ZmU6otyJWUPKQhRKpP1NZyo6W5I0ZqfF4FKUl58TWqvJJnAmLeSSlJefE1qrySZwJi3En9391E7lba0+mfAbUQyxIV78M+AEQdJbCFDW+srClWXHQEeb7ijpANh/wPJKD6QpkwsQwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2 & 1\\\\0 & 2 & -2\\\\0 & 0 & 5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2  1 ⎤\n",
       "⎢        ⎥\n",
       "⎢0  2  -2⎥\n",
       "⎢        ⎥\n",
       "⎣0  0  5 ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L_inv * A # Later we'll call this result the matrix U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now have the following, shown in (19)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ {L}^{-1}{A}={U} \\tag{19} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Left-multiplying by $L$ leaves (20)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ {L}{L}^{-1}{A}={L}{U} \\tag{20} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inverse of a square matrix multiplied by itself gives the identity matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ {I}{A}={L}{U} \\\\ {A}={L}{U} \\tag{20} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can construct $L$ from $E32$ and $E21$ above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ {E}_{21}^{-1}{E}_{32}^{-1}{E}_{32}{E}_{21}={E}_{21}^{-1}{E}_{32}^{-1}{U} \\\\ \\therefore {E}_{21}^{-1}{E}_{32}^{-1}={L} \\tag{21} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\3 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢3  1  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  0  1⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E21.inv() # The inverse is easy to understand in words\n",
    "# We just want to add 3 instead of subtracting 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAYAAAD010ABAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFbklEQVR4Ae2dX27UMBDGF8QzqkBCvLK9QaEnaLkBhRMANwDx1L6h9ga0J0BwA8oJgL0BPCMkyooT8H2pp7ipnZ3sZPJHtSXXjuOd8fwy8cbJbHpjf39/azabfUNOpY8HBwd7qR2l7ZwA+HxHbZ7igX03bkU7jlFn5zj9iDdKPUngEK0btT3b2H7CthjwIYgXoDVSqzbBjI55KaGNcK8AvtRpbBsYND3lN/ISeRP5PdoWKN2TRffNtqPj0UH+g1w/LdqKUveHLn5HfEF5hEyPeYv8AfXk3KcWrOho1R1PEVl1UEKYJ8icQmhUn3BfUCfG8BFllVBfhu0PaHgYmjsvoMOsW+XBwaA9lK9hxafOLWkWyKuY00SXL2jbwpg8D7ZZtwpwwrg+m3YzypahPbc/87FWzTnZat2jBhx551kDFk5ZnaeudI8aMKjdCeTEY2KQTdDjfuvWO9E9dsACp2mevSudnEqT7rEDbvJS8TBeG3ukTnSPGjDmQZkaBGYMUjzLZfXZle5RAw40eYkmMGPAAj11CRf3s9TNuqcAmIuJRwlKXC4vIk9LdDE3mXWvAzjlTWZLcgIAkEvjM5TVzRP2Q51j4CrrObe9Uhe6VUtlGgBl71DwtJSL789o+4rtb2EgqLolLoffQM88aODtwB1sL9w0/hds0t0G8Mv/OvutAeQSGrlM7z1Zda8zRfRu5JQVFsDOR68ALoCdCTiLLx5cADsTcBZfPLgAdibgLL54cAHsTMBZvHqpzHFg2ThY8Iczh0bxFrvVUwSUDBb8IdZjDJMLelF5MAwzB2AIpLYldPPW5GSDXrQebA7AaAtW+gMwo3gmG/SiBSz3gMVuKXkbkSm3/3zvdP/m7FLbvRJwOEWJqOkp63y6DNMj78rulYChXh4uylGLR9QEPe43xXondmsACxx+2eSSd/BHTm8f7Sa7NYCbvFSOslfwRx8Aczo6sZuAbyP/CuUVZfwWD40CM+4jR9cl+CNW1HfdaPd9jJdMZwT8F/leKFEkkzkAIyl1/I3r2v0TppFpBVhjpjkAQ6NkhH3Mdmvm4BlOl8GCP2rQZUqqNftsdmG3aqkchm8KwLAggKGTDXpRA4aR/LIbKvhjskEvqinC4n3X/bMFsLMHFMAFsDMBZ/HFgwtgZwLO4osHF8DOBJzFFw8ugJ0JOItXL5U5DiyXBwk8gV7e5KFupvl5MXuN9kWoj7ZQA4YxDDx5i7J6MUYwmr8weozsdsM96DlBydCBKqFO2KLb84eIQWPlXPwZGeMzHkA/78uokmoOhsBk4Ak0EDbvmXomwrz0eziMhzedaKSrbujZQOaraziGZ8g8k1olFWBIHCzwBLqfIvPsqafqaQOMlymjvt+8DdnmoBct4N3MaOVUye3PfKxVc/WeIBib855ceyslXp1XzsGRYU1PWT29iDf6U2krNLrN/ymlbds0HixPk8VbYx1N0ON+ndZx0HnG8KDyNV+pcXWqzyJMA1jkN52KfQee8BES36s5yBMWAaIpNYCbvFS8u7fAE0Al3FOUF5dtGkOH6rMScHQKCsx4rOLVvcyDGMsrKkc52DO62HhNfSXgIGTdAAzNGFR9AJUX+psxXNT5RsC5SsBAnbSAzQEYFvsAkVcM2zHcIO/iJR0W+Z6fXXmZRuUw7BiZa3/+RiJeKnOFt+M5QOijh/IA80tN7kdwauKUxVcrHqHsI8l02EqXCnCQOFTgCeEScjX/1qxzn/txAE1BL2rAULSEcb1fFkFvbqFRY+2zCf2mL1TtHOwz+msgtQB2PsgFcAHsTMBZfPFgZ8DxVcR3fGPW1ZV/VFInUtsGs+w/KmFXAua1ZO7ya/QPFWnEwImLn+wi5B+h6/aT4DxX/wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  1  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  2  1⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E32.inv()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\3 & 1 & 0\\\\0 & 2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢3  1  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  2  1⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E21.inv() * E32.inv()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is exactly the inverse of our inverse of $L$ above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\3 & 1 & 0\\\\0 & 2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢3  1  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  2  1⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L_inv.inv()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is called _LU-decomposition_ of $A$.  More about this in two chapters from now (I_05_LU_decomposition)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As an aside we can also do elementary column operation, but then we have to multiply on the right of $A$ and not on the left as above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Example problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example problem 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve the linear system in (22)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} x-y-z+u &= 0 \\\\ 2x+2z &= 8 \\\\ -y-2z &= -8 \\\\ 3x-3y-2z+4u &= 7 \\end{align} \\tag{22} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -1 & 1 & 0\\\\2 & 0 & 2 & 0 & 8\\\\0 & -1 & -2 & 0 & -8\\\\3 & -3 & -2 & 4 & 7\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  -1  -1  1  0 ⎤\n",
       "⎢                ⎥\n",
       "⎢2  0   2   0  8 ⎥\n",
       "⎢                ⎥\n",
       "⎢0  -1  -2  0  -8⎥\n",
       "⎢                ⎥\n",
       "⎣3  -3  -2  4  7 ⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_augm = Matrix([[1, -1, -1, 1, 0], [2, 0, 2, 0, 8], [0, -1, -2, 0, -8], [3, -3, -2, 4, 7]])\n",
    "A_augm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & 0 & 0 & 0 & 1\\\\0 & 1 & 0 & 0 & 2\\\\0 & 0 & 1 & 0 & 3\\\\0 & 0 & 0 & 1 & 4\\end{matrix}\\right], \\  \\left( 0, \\  1, \\  2, \\  3\\right)\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  0  0  0  1⎤              ⎞\n",
       "⎜⎢             ⎥              ⎟\n",
       "⎜⎢0  1  0  0  2⎥              ⎟\n",
       "⎜⎢             ⎥, (0, 1, 2, 3)⎟\n",
       "⎜⎢0  0  1  0  3⎥              ⎟\n",
       "⎜⎢             ⎥              ⎟\n",
       "⎝⎣0  0  0  1  4⎦              ⎠"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_augm.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Whoa! That was easy!  Let's take it a notch down and create some elementary matrices.  First off, we want the matrix of coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -1 & 1\\\\2 & 0 & 2 & 0\\\\0 & -1 & -2 & 0\\\\3 & -3 & -2 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  -1  -1  1⎤\n",
       "⎢            ⎥\n",
       "⎢2  0   2   0⎥\n",
       "⎢            ⎥\n",
       "⎢0  -1  -2  0⎥\n",
       "⎢            ⎥\n",
       "⎣3  -3  -2  4⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, -1, -1, 1], [2, 0, 2, 0], [0, -1, -2, 0], [3, -3, -2, 4]])\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we need to get rid of the $2$ in position row `2`, column `1`.  We start by numbering the elementary matrix by this position and modifying the identity matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -1 & 1\\\\0 & 2 & 4 & -2\\\\0 & -1 & -2 & 0\\\\3 & -3 & -2 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  -1  -1  1 ⎤\n",
       "⎢             ⎥\n",
       "⎢0  2   4   -2⎥\n",
       "⎢             ⎥\n",
       "⎢0  -1  -2  0 ⎥\n",
       "⎢             ⎥\n",
       "⎣3  -3  -2  4 ⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E21 = Matrix([[1, 0, 0, 0], [-2, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])\n",
    "E21 * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now for position row `3`, column `2`.  We have to use row `2` to do this.  If we used row `1`, we would introduce a non-zero into position row `3`, column `1`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -1 & 1\\\\0 & 2 & 4 & -2\\\\0 & 0 & 0 & -1\\\\3 & -3 & -2 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  -1  -1  1 ⎤\n",
       "⎢             ⎥\n",
       "⎢0  2   4   -2⎥\n",
       "⎢             ⎥\n",
       "⎢0  0   0   -1⎥\n",
       "⎢             ⎥\n",
       "⎣3  -3  -2  4 ⎦"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E32 = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, Rational(1, 2), 1, 0], [0, 0, 0, 1]])\n",
    "E32 * (E21 * A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now for the $3$ in position row `4`, column `1`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -1 & 1\\\\0 & 2 & 4 & -2\\\\0 & 0 & 0 & -1\\\\0 & 0 & 1 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  -1  -1  1 ⎤\n",
       "⎢             ⎥\n",
       "⎢0  2   4   -2⎥\n",
       "⎢             ⎥\n",
       "⎢0  0   0   -1⎥\n",
       "⎢             ⎥\n",
       "⎣0  0   1   1 ⎦"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E41 = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [-3, 0, 0, 1]])\n",
    "E41 * (E32 * E21 * A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's exchange rows `3` and `4`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & -1 & -1 & 1\\\\0 & 2 & 4 & -2\\\\0 & 0 & 1 & 1\\\\0 & 0 & 0 & -1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  -1  -1  1 ⎤\n",
       "⎢             ⎥\n",
       "⎢0  2   4   -2⎥\n",
       "⎢             ⎥\n",
       "⎢0  0   1   1 ⎥\n",
       "⎢             ⎥\n",
       "⎣0  0   0   -1⎦"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ee34 = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]])\n",
    "Ee34 * E41 * E32 * E21 * A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see where that leaves $\\mathbf{b}$.  After all, what we do to the left, we must do to the right."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ {Ee}_{34}\\times{E}_{41}\\times{E}_{32}\\times{E}_{21}{A}{x}={Ee}_{34}\\times{E}_{41}\\times{E}_{32}\\times{E}_{21}{b} \\tag{23}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\8\\\\-8\\\\7\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0 ⎤\n",
       "⎢  ⎥\n",
       "⎢8 ⎥\n",
       "⎢  ⎥\n",
       "⎢-8⎥\n",
       "⎢  ⎥\n",
       "⎣7 ⎦"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_vect = Matrix([[0], [8], [-8], [7]])\n",
    "b_vect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\8\\\\7\\\\-4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0 ⎤\n",
       "⎢  ⎥\n",
       "⎢8 ⎥\n",
       "⎢  ⎥\n",
       "⎢7 ⎥\n",
       "⎢  ⎥\n",
       "⎣-4⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ee34 * E41 * E32 * E21 * b_vect"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's print them next to each other on the screen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1 & -1 & -1 & 1\\\\0 & 2 & 4 & -2\\\\0 & 0 & 1 & 1\\\\0 & 0 & 0 & -1\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\\\8\\\\7\\\\-4\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡1  -1  -1  1 ⎤  ⎡0 ⎤⎞\n",
       "⎜⎢             ⎥  ⎢  ⎥⎟\n",
       "⎜⎢0  2   4   -2⎥  ⎢8 ⎥⎟\n",
       "⎜⎢             ⎥, ⎢  ⎥⎟\n",
       "⎜⎢0  0   1   1 ⎥  ⎢7 ⎥⎟\n",
       "⎜⎢             ⎥  ⎢  ⎥⎟\n",
       "⎝⎣0  0   0   -1⎦  ⎣-4⎦⎠"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ee34 * E41 * E32 * E21 * A, Ee34 * E41 * E32 * E21 * b_vect"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we can simply do back substitution.  We note that $-1u = -4$ and thus $u = 4$.  From here, we work our way back up."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "$$ \\begin{align} -1(u) = -4 \\quad &\\therefore \\quad u=4 \\\\ 1(z)+1(4) = 7 \\quad &\\therefore \\quad z=3 \\\\  2(y) + 4(3) - 2(4) = 8 \\quad &\\therefore \\quad y=2 \\\\ 1(x)-1(2)-1(3)+1(4)=0 \\quad &\\therefore \\quad x=1 \\end{align} \\tag{24}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}